Info2 - Euclid Handout
Euclid of Alexandria (Greek : Εὐκλείδης, Eukleides) (* circa 365; † 275 BC) was a Greek mathematician, now known as “the father of geometry”. His most famous work is Elements, widely considered to be history’s most successful textbook. Within it, the properties of geometrical objects and integers are deduced from a small set of axioms, thereby anticipating (and partly inspiring) the axiomatic method of modern mathematics.1
Elements: Book VII, Proposition 2:
To find the greatest common measure of two given numbers not relatively prime.
Let AB and CD be the two given numbers not relatively prime. It is required to find the greatest common measure of AB and CD.
If now CD measures AB, since it also measures itself, then CD is a common measure of CD and AB. And it is manifest that it is also the greatest, for no greater number than CD measures CD.
But, if CD does not measure AB, then, when the less of the numbers AB and CD being continually subtracted from the greater, some number is left which measures the one before it. For a unit is not left, otherwise AB and CD would be relatively prime, which is contrary to the hypothesis.
Therefore some number is left which measures the one before it.
Now let CD, measuring BE, leave EA less than itself, let EA, measuring DF, leave FC less than itself, and let CF measure AE. Since then, CF measures AE, and AE measures DF, therefore CF also measures DF. But it measures itself, therefore it also measures the whole CD.
But CD measures BE, therefore CF also measures BE. And it also measures EA, therefore it measures the whole BA. But it also measures CD, therefore CF measures AB and CD. Therefore CF is a common measure of AB and CD.
I say next that it is also the greatest.
If CF is not the greatest common measure of AB and CD, then some number G, which is greater than CF, measures the numbers AB and CD.
Now, since G measures CD, and CD measures BE, therefore G also measures BE. But it also measures the whole BA, therefore it measures the remainder AE.
But AE measures DF, therefore G also measures DF. And it measures the whole DC, therefore it also measures the remainder CF, that is, the greater measures the less, which is impossible.
Therefore no number which is greater than CF measures the numbers AB and CD. Therefore CF is the greatest common measure of AB and CD.
Corollary